How to Use Sine Substitution to Integrate

With the trigonometric substitution method, you can do integrals containing radicals of certain forms because they match up with trigonometric functions. A sine can take the place of a radical in a particular form.

1. Draw a right triangle where

   

   You should confirm this with the Pythagorean theorem.

   
   

2. Solve

   

   differentiate, and solve for dx.

   

3. Find which trig function equals the radical over the a, and then solve for the radical.

   Look at the triangle in the figure.

   

4. Use the results from Steps 2 and 3 to make substitutions in the original problem and then integrate.

   Note that in this particular problem, you have to make three substitutions, not just two like in the first example.

   

5. The triangle shows that

   

   Now, substitute back for your final answer.

   

It’s a walk in the park.